Let us make some remarks at this point. First, \(\times \wr ~I_n\)  has a faithful representation by complex generalized permutation matrices. These are matrices with a single non-zero complex number in every row and column, or, equivalently, products \(DP\)  with \(D\)  a diagonal matrix without zeros on the diagonal and \(P\)  a standard permutation matrix. Given the conjugating nature of \(\Xi \) , the appearance of \(\times \wr ~I_n\)  is natural, as the generalized permutation matrices form the stabilizer of the diagonal matrices. Let us also refer to some earlier reports on the wreath product in combination with this parametrization. For example, in [1]} there is firstly a quotient by scaling and secondly the quotient by permutations. Although not reported as such, we recognize the wreath product there. In [2]}, it is shown that a restriction of \(\Xi \)  defines a principal bundle for a wreath product group, where the number field need not be \(. As we will argue now, the parametrization \)\( also defines a principal \) In\(-bundle. Let us use that a projection is a principal bundle if it is given by the quotient of a free and proper action. The latter is readily verified, hence it remains to be shown that \)\( coincides with this projection.\begin{Lemma} There is a principal bundle\begin{equation}\begin{tikzcd}\times \wr I_n {r} & \mathrm {Fr}(V)\times {\underline{n}}{r}{\Xi }& N(V).\end{tikzcd}\end{equation}\end{Lemma}\begin{proof}As stated, it suffices to show that \Xi  and the action quotient map are isomorphic as bundle maps. As we already saw that the action preserves the fibers of \Xi , it only remains to show that each group orbit exhausts the fiber in which it lies. Therefore, assume \Xi (\tilde{f},\tilde{\lambda })=\Xi (\tilde{f}^{\prime },\tilde{\lambda }^{\prime })=:A. Then both \tilde{\lambda } and \tilde{\lambda }^{\prime } constitute \mathrm {Spec}(A), hence they are related by a permutation \sigma \in S_n as \tilde{\lambda }=\sigma \tilde{\lambda }^{\prime }. Then f_i needs to be parallel to f_{\sigma ^{-1}(i)}^{\prime } for each i, so \tilde{f} and \sigma \tilde{f}^{\prime } are equal up to an element of (\times )^n. However, then (\tilde{f},\tilde{\lambda }) and (\tilde{f}^{\prime },\tilde{\lambda }^{\prime }) differ only up to an element of \times \wr I_n.\end{proof}\) This result has various consequences. First, it shows that \(N(V)\)  can be realized as a quotient space. Another important observation concerns the existence of local sections of \(\Xi \) . These sections provide local moving eigenframes and corresponding smooth eigenvalues, as we will use in Sec. . We state the details in the following corollaries.
